Subject: Geometry
Created by: Ariel Li
Revised: 03/29/2018
Basic Geometry Rules
Triangle
Definition: Triangles are closed geometrical figures that have three straight sides. Every triangle
will, as a result, have three angles as well.
The sum of the three angles in a triangle is always equal to 180 degrees (for definition of degree
see subsection below on angles)
Special Triangles:
A. The Right Triangle: The right triangle is a triangle that has one 90o angle. Since the sum of the
angles in a triangle must be 180o, this implies that the other two angles in a right triangle must add
up to 90o. One of these relations is the so called Pythagorean Theorem. For the right triangle shown
in figure 2, the relation is:
a2 + b2 = c2
B. Isosceles Triangle: In this special type of triangle, two sides are equal to each other. One can
prove that the angles opposite to these two equal sides are also equal to each other. Thus, let us
say that we have a triangle in which two sides are 5.0 cm in length and the other side is - let's say -
2.0 cm in length, as in the figure below. This means that the two angles indicated with an arrow in
the figure are equal.
Subject: Geometry
Created by: Ariel Li
Revised: 03/29/2018
Basic Geometry Rules
B. Equilateral Triangle: This is a special type of isosceles triangle in which not just two, but all
three sides, and as a result all three angles are equal. Since the sum of the angles in a triangle must
be 180o, this means that for an equilateral triangle each angle is 60o.
Perimeter: The perimeter of a triangle is equal to the sum of the length of its three sides.
Area: The area of a triangle is equal to half its base times its height .
The height of a triangle is each of the perpendicular lines drawn from one vertex to the
opposite side (or its extension).
Figure 5: Sample Triangle
Perimeter for Figure 5: P = a + b +c
where: P = Perimeter, a, b, c = (length of) sides
Area for Figure 5: A= ½ (h*b)
where: A = Area, b = (length of) base, h = (length of) height
Subject: Geometry
Created by: Ariel Li
Revised: 03/29/2018
Basic Geometry Rules
Circle
Definition: A circle is a closed geometric figure as shown in the following figure:
It is defined such that all the points on the circle are at a constant distance from a center. This
distance is called the radius of the circle, as indicated in the diagram above.
Circumference: The circumference of a circle depends very simply on the radius of the circle.
C=2×p×r
where: C = Circumference, r = radius, and p = 3.1415 ....
Area: The area of the circle also depends on the radius of the circle. It is given by:
A = p × r2
where: A = Area, r = radius, and p = 3.1415 ....
Parallelogram
Definition: a parallelogram is a quadrilateral with two pairs of parallel sides.
Figure 8: A Parallelogram
The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a
parallelogram are of equal measure.
Perimeter: The perimeter of a triangle equals the sum of the length of its four sides. In Figure 8,
P = a + b + a + b = 2a + 2b
where: a, b = length of sides
Area: The area of a triangle is equal its base times its height . The height of a triangle is
each of the perpendicular lines drawn from one vertex to the opposite side (or its
extension). In Figure 8,
A = b*h
where: A= area, b = base, h = height
Subject: Geometry
Created by: Ariel Li
Revised: 03/29/2018
Basic Geometry Rules
Trapezoid
Definition: A trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite sides
parallel.
Figure 9: A Trapezoid
Isosceles trapezoid is a trapezoid when the sides that aren't parallel are equal in length and both
angles coming from a parallel side are equal.
Perimeter: The perimeter of a trapezoid equals the sum of the length of all its four sides. In Figure
9,
P=a+b+c+d
where: a, b, c, d= length of sides
Area: The area of a trapezoid is equal half the sum of its base (or parallel sides) times
height. In Figure 9,
A = ½ * (a + b) * h
where: a, b = bases, h = height
Rectangle
Definition: A rectangle is a special parallelogram with straight sides where all interior angles are right
angles (90°).
Figure 10: A Rectangle
The opposite sides of a rectangle are parallel and of equal length. The longer sides are called
length and the shorter sides are called width.
Perimeter: The perimeter of a rectangle equals the sum of the length of all its four sides. In Figure
10,
P = l + w + l + w = 2l + 2w
where: l, w = length and width (four sides)
Area: The area of a rectangle equals its length times its width. In Figure 10,
A=l*w
where: l = length, w = width
Subject: Geometry
Created by: Ariel Li
Revised: 03/29/2018
Basic Geometry Rules
Square
Definition: A square is a special rectangular that all its four sides equal.
Figure 11: A Square
Perimeter: The perimeter of a square equals the sum of the length of all its four sides. In Figure 11,
P = a + a + a + a =4a
where: a = side
Area: The area of a square equals the length of the sides squared. In Figure 11,
A = a2
where: a = side
--
References - The following work was referenced to during the creation of this handout: The
University of Colorado Atlas Project, Analyze Math, Wolfram, Paul Bourke, and Trans4mind.
